Wave Optics

Chop
icple .Interferersce of Ligh
o f Liyht

na etectilieat
reflection of lghr and retraction
ined by corpslar theory of

somena likr
polariaion of lighr
ed b WE concegr of light called
pcal eptkca Huygrns Mathe fiss
waVe thery of g t () Cylindrical wavefront When the source of light
is ineat, cg a straight line source, slit etc as
the cxitence of hypochetical shown in the figure. All the points equidistant
ber for the popagation of wve Latet, from the source lie on a cylinder Therefore. the
ngspe med his famous interferencE wavefront is cylindrical in shape
fimally establshed that ight is
d we pheomena

yns Principle
ll he puicles of a medium, which
e sam pha Aline pergendiculat

shage of uce of light.

of thre tpes, which ane given
Plane wavetront When the poit souroe or linear
sphescal efnt When the sounce of light sOTGE t ght is at very large distance, A Sall
pone sooe the wefront is a sphere with portion of spherical or cylindrical wvefront appeans
te s the source
 384 12th

wl
to be plane. Such a wavefront is called a plane Note Huygens' argued that the amplitude of the secondary
wavelets is maXimum in the torward direction and
wavefront. F
in the backward direction. Hence, the backward
secondary wavefront is absent.
p
-Light rays Reflection and Refraction of Plane Waves at a
Plane
plane surface Using Huygens' Priniple
P
wavefronts
Plane wavefront Huygens' principle can be used to explain the
phenomena of reflection and refraction of light on the R
Hence, the wavefront is a surface of constant basis of wave theory of light.
phase. Reflection at aPlane Surface
Let 1, 2, 3 be the incident rays and 1', 2, 3 be the
Huygens' Principle corresponding reflected rays.
Inorder to explain how awavefront is propagated N
forwards through a homogeneous isotropic medium Incidenl
and toconstruct the position of a wavefront at any wavefront 2
2
instant of time, Huygens has given following two 3
assumptions. Reflected
wavefront
According to Huygens' principle,
() Each point on the given wavefront (called X
primary wavefront) is the source of a secondary Reflection of plane wave
disturbance (called secondary wavelets) and the Ifc is the speed of the light, is the time taken by light
wavelets emanating from these pointsspread out to go from B toC or A to D or E to G through F,
in all directions with the speed of the wave. then
The speed with which the wavefront moves EF FG
ourwards rom the source is called the speed of C
...(0)
the wave. The energy of the wave travels in a EF
direction perpendicular to the wavefront. In AAEF, sin i =
AF
(i) Asurface touching these secondary wavelets, FG
tangentially in the forward direction at any In AFGC, sin r =
instant of time gives the new wavefront at that FC
instant. This is called secondary wavefront. AF sin i FCsin r
or t=
C
G
ACsin r + AF (sin i - sin r)
t=
A Ag
B, (:: FC = AC - AF)
A
For rays of light from different parts on the incident
D,, Propagation of
light wave
wavefront, the values of AF are different. But light
from different points of the incident wavefront should
F, G
take the same time to reach the corresponding points
la
(-0 t=t on the reflected wavefront.
(b)
in Fig ial, EF, is the sechon of the given spherical So, rshould not depend upon AF.This is posble only. &
wavefront
and G, is the new wOvefront in the forward directhion Sin i- Sin r =0
In
Fig b). Fisthe secthon of the given plane wavefront and i.e.
GG, is the new wavefront in the forward direchon.
0r
Zi=Lr
 385

WNelh

law of releion
wvot , he relleing ...(iv)
lleed waveont CD eall
othe ple of he pper. vacuum, then
representsthe specd of lightin
eny, nomal to he miot NY and Now, il the refractive
Ie n he pleof the arc known as
and ,
wond av of relletion medium 2, respectivcly.
indices of medum Iand written as
Plono Surfoce Eg. (iv) can be
Relrotion (Snell'sLow) of o Intes of
rcfractive indices,
he incdent ays and I', 2. 3'be the
Hepondogthed tays
N

Sncll's law of refraction.

This is known as waveleng1hs of light in
2, denote the
Further, ifA,and 2, respcctively and if the
mcdum be
ncdun l and
cqual to A,, then the distance AD will
distance BC is
V cqual to , , thus
BC
le tod AD
wavehon\

kohocion of o plone wave

ghu
speeds of liçhe in the two ncdia and V, = V,
Il,,,arce the to Cor A to D
is lhetime taken by
lighe o go from B
the frequency does not change on refraction.
Hence, characteristic of the source,
hen a
o toGdhtogh F, Thus, frequency v bcing
light travels from one
medium to
remains the same as
another.
proportional to
Also, wavelength is directly proportional to
the (phase) speed and inversely
refractive index.
AF sin Csin (i)

denser
sini sin r undergoes refraction from
AC sin r Whenthe plane waveconditions remain same except
to rarer medium, all greater than the angle of
diflercnt parts on the incident that angle of refraction is
Tot tays of light from incidence and v, >:
different. But light then r=90°. (so. for
Wavehon, the values of AF ate (critical angle),
dilterentpoints of the incident wavefront should " Also, when i i
greater than cirtical angle, no
utm coesponding points all angles of incidence
take the same timetorcah the undergo total internal
should not depend wave is rctracted and it
o the etrated wavefront So, t reflection).
pon A. This is possible only, if
386

Behaviour of Prism, Lens ond Spherical Mirror rirror Tog

towards Plane VWavefront
) Behaviourof aprism Since, the speed of light Theo
waves ate less in glass, o the Iower portioi of the
1.
incoming wavefot (which travels through the Srsrca wavetrort of radius P/2
getest thickness of glass prissn) will gn deiayed Pafoctin of pione wove by concave mirror 2
resulting in a tit in the erierging wavefront.

EXAMPLEl. Aplare wavefront is incident from air (u =

at an angle of 37 with a horizontal
boundary of a
3
refractive medium frorn air of refractive index =. Find
2

the angle of refracted wavefront with the horizontal

broundary.
Sol It has been given that incident wavefront makes 37°
(6) Behavigur of alens The central patof the with horiontal. Hence, incident ray makes 53" with
incident plane wave travetes thethickest puttion normal as the ray is perpendicular to the wavefront.
sf the lens and is delayed the mst. Incidert
Wavsfront
iniiert fay
Due to this, the eserging wavefont has a
depressin at the entre. Therefore, the wavefront
tecones spherical and nveryes to the point F 37
which is knwn as the frus.
Refracted
wavotrort
Relracted

sin 53"
iun, by Snell's law,
2
( ) Behaviour of a spherical mior The entral part sinr sin 53
3
of thr iticiient wavefonr travels the largest
disair beite refiection fron the cCaVE rsin (0660.79)
mitut Hece, gets delayed, as a esult of which 3143
the sefieced waveist is spherical whch whhis saie the as angle f refractive wavelont with
tixergs ar the frxcal int I honnal
 Interference of Light
Coherent and Incoherent Sources Two cohcICHt sources can be obtaned cither
Light sources are of two types, i.e. coherent and (a) the source and its virtual image (loyd's i
non-coherent light sOurceS. (b) the Wo virtual images ol he same soure
The sources of light which enmit light waves of same (Fresnl's biprism)
() (wo real imagcs of the same sourCe (Younp's Frin
wavelength, same frequency and are in same phase or double sli).
having constant phase difference are known as
coherent sources. (ii) The two sources should give monodhromatic lod, wid

Two such sources of light, which do not emit light waves (iii) The path difference betwecn liglht waves from tw.
sources should be small.
with constant phase difference are called incoherent
sOurces.
As discussed earlier, when two
independent sources of Young's Double Slit Experiment Int
light emits monochromatic waves of Young in 1802,experimentally demonstrated the
and phase difference meet at a point,intensities
then the
/, ,
phenomenon of interterence of light. The set-up is
resultant intensity is given by shown in figure given below
|I=1, +l, +2/1,1, cosQ
Here, the term 2//, 1, cos is called interference
term. There are two possibilities.
() If cos remains constant with time, then the total
intensity at any point will be constant.
The intensity will be maximum (|1, t,)' at
points, where cos) is 1 Young's double slit arrangement lo produce interference patern
and minimum (//, -1,)' at point, where cos Suppose S, and S, are two fine slits, a small distanced
is -1. apart. They are illuminated by a strong source S of
Hence, to obtain interference,we need two monochromatic light of
at a distance D from the wavelengh . MN is a screen
sources with same frequency and with a constant slits.
phase difference. Forconstructive interference (Bright
The two sources in this case are
coherent sources. The distance of nth bright fringe fromfringes)
() If cos¢varies continuously with time nD.
point 0 is
both positive and negative value, then assuming
the average
y= |where, n =0, 1, 2,3, ..!
value of cos over a full cycle will be zero. Hence, forn =0, y, = 0at Ocentral bright
The interference term averages to zero. D. fringe
be same intensity, / =I +l, at There will for n=1, y, =or Ist bright fringe
Hence, the two sources in this caseevery point.
are incoherent
SOurces. for n=2, 2D.
d
for 2nd bright fringe
Conditions for Obtaining Two for n =n, nDA
Coherent Sources of Light y, = for nth briglht fringe
Following are the requirements (conditions) for For destructive
interference (Dark
obtaining two coherent sources of light Thedistance of nh dark fringe fromfringes)
) Coherent sources of
light should be obtained DÀ
point Ois,
from asingle source by some y= (2n-) ,|where, n 12, 3..1
device.
 Wave Optics

Hence. for n =1 y/ = for Ist dark fringe EX

2d th
for n =2, 3DÀ
= for 2nd dark tringe
2d

for n=, =(2n - I) DA for ah dark

2d fringe
Fringe width The separation berween any two
consecutive bright or dark tringes is called fringe
fro. width. It is given by
B=
D.
d

Intensity of the Fringes

Pis For a bright fringe, o =2uit EX
and coso = cos 2rT = 1 frir
So, W
I=1mux
=l,+l, +21,/, =4/,
So
(as, I, = 1, =l,in YDSE)
. Intensiy of a bright fringe = 4l, = constant
For a dark fringe, o = (2n - )
cosO =-1
So,

=1, +I, -2\1,I, =0

.. Intensity of a dark fringe = 0 EXA
The intensity at any point is S'su
on
I=41, cos (0/2) the

Note if YDSE apparatus is immersed in a liquid of refractive

index u, then wavelength of light and hence fringe
width decreases u times.

Distribution of Intensity
The distribution of intensity in Young's double slit So
experiment is shown below
4I, (Intensity with rvo coherent sources)

21Untensiry with rwo incoherent surces)

ntensir with only one source)

-24 - 3À2 -À -2 N2 à 3/2 22 - Path difference

- 4 -3R -27 - 27 3r dn - Phase ditference

Hence, all bright interference fringes have same

intensity 4I, and all dark fringes have zero intensity.
 Conditions for Sustained Interference
In order to obtain a well defined observable
interference pattern, the intensity at points of
constructive and destructive interference must be
maintained maximum and almost zero, respectively.
For this, following conditions must be satisfied
() The two sources of light producing interference
must be coherent.
s (ii) The two interfering waves must have the same
plane of polarisation.
(iii) The two sources must be very close to each other
and the pattern must be observed at a larger
distance to have sufficient width of the fringe

()
(iv) The sources must be monochromatic, otherwise
the fringes of different colours will overlap.
(v) The twowaves must be having same amplitude
for better contrast between bright and dark
fringes.
 DiffroctionofLight
single wavefront rcach the
The wavelets from the in same phase. Hence,
phenomenon of bending of light around the sharp centre C on the screen
The spreading of light within the geometrical givecentral maximum
cornersandthe interfere constructively to
shadowofthe opaque
obstacles is called diffraction of light. (bright fringe).
thus deviates from its linear path. The obtained on thescreen
Thelight becomes much more pronounced, when the Thediffraction pattern band, having alternate
deviation comparable consists of acentral bright decreasing intensity
dimensions of the aperture or the obstacle are dark and weak bright bands of
oflight.
ro the wavelength
on both sides.
Diffracted
Diffracted Incident wave
Incident Wave wave
wave

a a

Bearn of | M
M
Light from source a MT. " M,
N
Screenl a =1
a>
Diffracted
wave

Incident
wave
Geomelry of single slit
diffraction

secondary minima, the slit can be divided

for nth
into2n equal parts.
a<
different width
Hence, for th secondary minima,
Diffraction of waves for slits of =
Path difference = sin
2 2

Diffraction of Light at a Single Slit is made

n2.
, (n=1, 2, 3, ...)
light with a plane wavefront or sin ,=
Aparallel beam of As width of the slit LN= a is
single slit LN. slitcan be divided
for nth secondary maxima, the
to fall on a diffraction
the order of wavelength of light, therefore
of through the slit. into (2n + 1)equal parts.
ocurs when beam of light passes
 From If Foreither It minima.
0 Wid1h
mathenmatical are
is diffractionmaxima.T
Eqs. is first he shown
below as
the Hence, or
small, side secondary point Hence,
secondarya
(i) sin distance of
and sin b= ofCentral And
the C the
D maxima
conditions a for
(11), corresponds
the diffraction sin sin
central
between minima. mh
minimum, , ,
we or Maximum Path
andposition secondary
==(2n+
D=
or= get sin = difference -2 (2n+
bright first minima
for The
to pattern
-32,-2h,-2, C 1) 1)
secondary interference above the 2a maxima,
fringe (d Intensity()
D. are position
sin can
conditions
exactly ) [n=
C 22 be
minimum maxima graphically 12,
h,of
reverse22.,3... 32 3,
...(1) ...) for central ...]
X
and
on of

(iv) (iii) (i) between

Following decreases.
Difference maximum As,
"m).(z (i)
hardly
ilable.
common
Therefore, Diffraction minima
10al betweenwhercas, Interference
contrastusuallyunequal.
always
In width.successive
fringes
bright fringes same
In
the wavefront. of the
secondarywhereas two The
types interference interference
bright Angular
width Width
coherent slit
superposition
interference Between
are
is in zero Whereas, is widh
as ofbright between
never same. diffraction the
diffraction wavelets fringes of
waves. isdiffraction
a or fringes central
pertures pattern, Interference
sources important
general and zero pattern, very Whereas, and increases, z
bright the are of
For dark small may coming is of central maximum
and width not diffraction
is
light
ofvisible
the gocs the th e
visiblecharacteristic pattern, or of in called points and width
fringes. there and and of same superposition
intensity
diffraction theof waves
may on from Diffractionmaxima.
light, darkintensity =
of isthe there diffraction
decreasing interference. of of of 2y=
light a not intensity different coming difference central
this of light. 20
à intensity
poor fringes, is of be 2D.
is
size not exhibited is good pattern, al of =
very contrasi of and bright points
minima fringes sameintcnsi from Liaht
are so f
smal the al